Dynamics Of Quasi-periodic Cocycles And Applications

SL(2,R)cocycle is an important research object in dynamical systems.Due to its rich dynamical properties and closely connection to operator spectral theory,SL(2,R)cocycle has attracted the attention of researchers.In this thesis,we are dedicated to study the dynamics of SL(2,R)cocycles and the spectrum of the extended Harper's model.In the first chapter,we give a brief introduction to the dynamics of cocycles including the definitions of Lyapunov exponent,fiber rotation number,the reducibility and hyperbolicity.And we introduce basic spectral theory of quasi-periodic Jacobi operator.Then we introduce some basic connections between the dynamics of cocycles and spectral behaviors of Jacobi operator.In the second chapter,we present a new application of the fast approximation method introduced by Anosov and Katok.By the constructions,we study the dynamics of quasi-periodic SL(2,R)cocycles from three aspects:the regularity of Lyapunov exponent,the growth of cocycles and the density of Cs reducible cocycles.In the third chapter,for non-critical extended Harper's model with Diophantine frequency,we establish the exponential decay of the upper bounds on the spectral gaps and prove the spectrum is homogeneous based on Avila's almost reducibility theorem.Especially we proved that the decaying rate is close to Lyapunov exponent in non-self-dual region.